Question
\( \begin{array}{lll}\text { d } x^{2}-6 x=-3 & \text { e } x^{2}+10 x=1 & \text { f } x^{2}-8 x=5 \\ \text { g } x^{2}+12 x=13 & \text { h } x^{2}+5 x=-2 & \text { i } x^{2}-7 x=4\end{array} \)
Ask by Johnson Pena. in China
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**d) \( x = 3 + \sqrt{6} \) or \( x = 3 - \sqrt{6} \)**
**e) \( x = -5 + \sqrt{26} \) or \( x = -5 - \sqrt{26} \)**
**f) \( x = 4 + \sqrt{21} \) or \( x = 4 - \sqrt{21} \)**
**g) \( x = 1 \) or \( x = -13 \)**
**h) \( x = \frac{-5 + \sqrt{17}}{2} \) or \( x = \frac{-5 - \sqrt{17}}{2} \)**
**i) \( x = \frac{7 + \sqrt{65}}{2} \) or \( x = \frac{7 - \sqrt{65}}{2} \)**
Solution
**d) Solve \( x^2 - 6x = -3 \)**
1. Write the equation in standard form:
\[
x^2 - 6x + 3 = 0.
\]
2. Identify the coefficients: \( a=1 \), \( b=-6 \), \( c=3 \).
3. Compute the discriminant:
\[
\Delta = b^2 - 4ac = (-6)^2 - 4(1)(3) = 36 - 12 = 24.
\]
4. Apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{6 \pm \sqrt{24}}{2}.
\]
5. Simplify the square root:
\[
\sqrt{24} = 2\sqrt{6}, \quad \text{thus } x = \frac{6 \pm 2\sqrt{6}}{2} = 3 \pm \sqrt{6}.
\]
---
**e) Solve \( x^2 + 10x = 1 \)**
1. Write the equation in standard form:
\[
x^2 + 10x - 1 = 0.
\]
2. Identify the coefficients: \( a=1 \), \( b=10 \), \( c=-1 \).
3. Compute the discriminant:
\[
\Delta = 10^2 - 4(1)(-1) = 100 + 4 = 104.
\]
4. Apply the quadratic formula:
\[
x = \frac{-10 \pm \sqrt{104}}{2}.
\]
5. Simplify the square root:
\[
\sqrt{104} = 2\sqrt{26}, \quad \text{thus } x = \frac{-10 \pm 2\sqrt{26}}{2} = -5 \pm \sqrt{26}.
\]
---
**f) Solve \( x^2 - 8x = 5 \)**
1. Write the equation in standard form:
\[
x^2 - 8x - 5 = 0.
\]
2. Identify the coefficients: \( a=1 \), \( b=-8 \), \( c=-5 \).
3. Compute the discriminant:
\[
\Delta = (-8)^2 - 4(1)(-5) = 64 + 20 = 84.
\]
4. Apply the quadratic formula:
\[
x = \frac{8 \pm \sqrt{84}}{2}.
\]
5. Simplify the square root:
\[
\sqrt{84} = 2\sqrt{21}, \quad \text{thus } x = \frac{8 \pm 2\sqrt{21}}{2} = 4 \pm \sqrt{21}.
\]
---
**g) Solve \( x^2 + 12x = 13 \)**
1. Write the equation in standard form:
\[
x^2 + 12x - 13 = 0.
\]
2. Identify the coefficients: \( a=1 \), \( b=12 \), \( c=-13 \).
3. Compute the discriminant:
\[
\Delta = 12^2 - 4(1)(-13) = 144 + 52 = 196.
\]
4. Apply the quadratic formula:
\[
x = \frac{-12 \pm \sqrt{196}}{2} = \frac{-12 \pm 14}{2}.
\]
5. Find the two solutions:
\[
x = \frac{2}{2} = 1, \quad x = \frac{-26}{2} = -13.
\]
---
**h) Solve \( x^2 + 5x = -2 \)**
1. Write the equation in standard form:
\[
x^2 + 5x + 2 = 0.
\]
2. Identify the coefficients: \( a=1 \), \( b=5 \), \( c=2 \).
3. Compute the discriminant:
\[
\Delta = 5^2 - 4(1)(2) = 25 - 8 = 17.
\]
4. Apply the quadratic formula:
\[
x = \frac{-5 \pm \sqrt{17}}{2}.
\]
---
**i) Solve \( x^2 - 7x = 4 \)**
1. Write the equation in standard form:
\[
x^2 - 7x - 4 = 0.
\]
2. Identify the coefficients: \( a=1 \), \( b=-7 \), \( c=-4 \).
3. Compute the discriminant:
\[
\Delta = (-7)^2 - 4(1)(-4) = 49 + 16 = 65.
\]
4. Apply the quadratic formula:
\[
x = \frac{7 \pm \sqrt{65}}{2}.
\]
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Simplify this solution Bonus Knowledge
To solve each of these quadratic equations, you can rearrange them into standard form \( ax^2 + bx + c = 0 \) and then apply the quadratic formula or factoring methods. For example, equation \( \text{d} \) becomes \( x^2 - 6x + 3 = 0 \). You would then identify \( a = 1 \), \( b = -6 \), and \( c = 3 \). You can confidently solve these equations using various methods! A common mistake is forgetting to move all terms to one side of the equation before applying the quadratic formula or trying to factor. Remember, organizing your equations in standard form makes the solution process much simpler. Happy problem-solving!
